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04-BS-7 · May 2016

Question 9 of 13: Lift, Drag, and Thrust Power of a Cruising Boeing 747

Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)

Notes on this paper

04-BS-7 Mechanics of Fluids — May 2016 (National Examinations, three hours, closed book). Section A (Calculative) offers 9 questions and instructs "do seven"; Section B (Analytical/Graphical) offers 4 questions and instructs "do three." Every question is answered below (13 of 13), so students can use the full paper as a study resource. Constants used throughout (from the paper's own Constants page): g = 9.81 m/s², patm = 100 kPa (an atmospheric head of 10 m of water is specified separately for Question 1), ρwater = 1000 kg/m³, SGglycerine = 1.26, SGmercury = 13.56, ρconcrete = 2400 kg/m³, ρair = 1.19 kg/m³ (20°C) / 1.21 kg/m³ (15°C), μwater = 1.0×10⁻³ N·s/m², Rair = 287 J/kg·K.

Reference texts: F. M. White, Fluid Mechanics, 8th ed. (McGraw-Hill) — fluid statics and manometry (Ch. 2), hydrostatic forces and the middle-third rule (Ch. 2), dimensional analysis and drag (Ch. 5, 7), pipe friction and the Moody/Colebrook relation (Ch. 6), control-volume momentum (Ch. 3); B. R. Munson et al., Fundamentals of Fluid Mechanics — jets, orifices, and streamline patterns (Ch. 5, 8); J. D. Anderson, Fundamentals of Aerodynamics — wave/compressibility drag divergence (Ch. 5) for the Boeing 747 wind-tunnel chart used in Question 9.

Question 9: Lift, Drag, and Thrust Power of a Cruising Boeing 747 (5 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given.

QuantityValue
Aircraft weight (320 Mg)3139.2 kN
Wing area S511 m²
Altitude conditionsT = −50°C = 223.15 K, p = 26 kPa
Mach number0.89
Gas constant R (air) / k287 J/kg·K / 1.4

Find. (a) CL; (b) CD from the wind-tunnel chart (extended-cab curves); (c) thrust power.

Approach. Get air density from the ideal gas law, speed of sound and true airspeed from the Mach number, then CL from steady level flight (lift = weight); read CD off the extended-cab (dashed) wind-tunnel curves at that CL and Mach number; finally thrust = drag in steady flight, so power = drag × velocity.

  1. Air density and speed of sound at altitude. $$\rho = \frac{p}{RT} = \frac{26\,000}{287(223.15)} = 0.4060\ \text{kg/m}^3, \qquad a=\sqrt{kRT}=\sqrt{1.4(287)(223.15)}=299.4\ \text{m/s}$$
  2. True airspeed and dynamic pressure. $$V = 0.89(299.4) = 266.5\ \text{m/s}, \qquad q=\tfrac12\rho V^2 = \tfrac12(0.4060)(266.5^2) = 14\,416\ \text{Pa}$$
  3. (a) Lift coefficient (L = W in steady cruise). $$C_L = \frac{W}{qS} = \frac{3\,139\,200}{14\,416(511)} = \boxed{0.426}$$
  4. (b) Drag coefficient, read from the extended-cab (dashed) curves at M = 0.89. Reading the dashed CL=0.4 and CL=0.5 curves at the M=0.89 gridline and interpolating linearly to CL=0.426 (marked on the reproduced chart below): $$C_D \approx 0.0234 + \frac{0.426-0.4}{0.5-0.4}(0.0301-0.0234) = \boxed{0.0252}$$
  5. (c) Thrust power (T = D in steady, level, constant-speed cruise). $$D = C_D\, q\, S = 0.0252(14\,416)(511) = 185.3\ \text{kN}$$ $$P = D\,V = 185\,300(266.5) = \boxed{49.4\ \text{MW}}$$

[Figure not reproduced: Boeing 747 CD-CL-Mach wind tunnel chart. See the official exam paper or the cited reference text.]

Fig. Q9 — original wind-tunnel chart (Attachments p.10). At M=0.89 the extended-cab (dashed) curves for CL=0.4 and CL=0.5 were read and linearly interpolated to the computed CL=0.426, giving CD≈0.0252.
QuantityResult
CL0.426
CD (extended cab, M=0.89)0.0252
Drag = required thrust185.3 kN
Thrust power49.4 MW